Finite Group Databasebeta

← #11921915

Subgroups — #11921915

Order 1152.

756 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×A4×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×C4×S4C211yesmaximal, characteristic
3C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4(C2×S4)⋊Dic3C212yesmaximal
5#10593238C212yesmaximal, complemented
6#85506—31nomaximal
7#82349—31nomaximal
8C22×C3×S4C2211yescharacteristic
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11A4×Dic6—22no—
12C2×C3⋊SL(2,ℤ/4)C2212yes—
13C2×C3×SL(2,ℤ/4)C2211yescharacteristic
14C3×C4×S4—22no—
15#63856—41nomaximal
16C23×Dic6D311yescharacteristic, complemented
17C2×Q8×A4—31no—
18C2×C4×S4D311yescharacteristic
19C2×C3×C4×D4—31no—
20C2×#924—31no—
21C2×#924—31no—
22(C2×S4)⋊C4—32no—
23C4⋊SL(2,ℤ/4)—31no—
24C4⋊SL(2,ℤ/4)—61no—
25SL(2,ℤ/4)⋊C4—32no—
26#5220—32no—
27#5212—31no—
28#4979—32no—
29#4947—32no—
30#4946—32no—
31#4929—31no—
32#4929—31no—
33C22×C3×A4C2311yescharacteristic
34C2×C3×S4—21no—
35C2×C3×S4—21no—
36C2×C3×S4Q812yescomplemented
37C2×C3×C4×D3—41no—
38C2×C3×Dic6—41no—
39C3×C4×A4D412yes—
40Dic3×A4—22no—
41C3⋊SL(2,ℤ/4)—22no—
42C3×SL(2,ℤ/4)—21no—
43C4⋊(C3⋊Dic3)—41no—
44#3724—42no—
45(C2×D3)⋊Dic3—42no—
46#2409—91noSylow
47C22×S4C2×D311yescharacteristic
48C22×C3×D4—31no—
49C23×C3×C4—31no—
50C23×C3×C4C2×D311yescharacteristic
51C23×Dic3C2×D312yes—
52C22×Dic6—22no—
53C22×Dic6—31no—
54C22×Dic6—31no—
55C22×Dic6—32no—
56C22×Dic6—62no—
57Q8×A4—62no—
58C2×C4×A4—21no—
59C2×C4×A4—32no—
60C2×C4×A4C2×D311yescharacteristic
61C2×SL(2,ℤ/4)—32no—
62C2×SL(2,ℤ/4)—62no—
63C2×SL(2,ℤ/4)C2×D311yescharacteristic
64C4×S4—22no—
65C3×C4×D4—62no—
66C3×C4×D4—62no—
67C2×C3×C4⋊C4—31no—
68C2×C3×C22⋊C4—31no—
69C2×C3×C22⋊C4—31no—
70C2×C3×C42—31no—
71C2×#217—32no—
72C2×#217—32no—
73C2×#217—32no—
74#946—121no—
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—31no—
77#924—64no—
78#924—61no—
79#924—62no—
80#924—61no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—32no—
83C3⋊#95—32no—
84C3⋊#95—32no—
85C3⋊#95—32no—
86C3⋊#95—32no—
87C3⋊#95—32no—
88C22×C3×D3—41no—
89C2×C3×A4C2×D411yescharacteristic
90C2×C3×A4C2×Q811yescharacteristic
91C2×C3×A4#4111yescharacteristic
92C3×S4—22no—
93C2×C32×C4—41no—
94C2×C3⋊Dic3—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—42no—
97C3×C4×D3—82no—
98C3×Dic6—82no—
99C23×Q8—31no—
100C2×C22⋊Q8—91no—
101C2×C22⋊Q8—91no—
102C2×C4×D4—31no—
103#395—92no—
104#393—92no—
105#391—91no—
106#391—91no—
107#386—92no—
108#385—92no—
109#380—91no—
110C24×C3—31no—
111C24×C3C22×D311yescharacteristic
112C22×A4—21no—
113C22×A4C22×D311yescharacteristic
114C2×S4—21no—
115C2×S4—21no—
116C2×S4Dic612yescomplemented
117C2×C3×Q8—121no—
118C2×C3×D4—32no—
119C2×C3×D4—32no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—32no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—62no—
132C22×C3×C4—62no—
133C22×C3×C4—61no—
134C22×C3×C4#6712yes—
135C22×Dic3—22no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—62no—
141C2×C4×D3—41no—
142C2×Dic6—34no—
143C2×Dic6—31no—
144C2×Dic6—32no—
145C2×Dic6—62no—
146C2×Dic6—62no—
147C2×Dic6—61no—
148C2×Dic6—64no—
149C2×Dic6—64no—
150C2×Dic6—62no—
151C2×Dic6S411yescharacteristic, complemented
152C4×A4—22no—
153C4×A4—62no—
154C4×A4#6712yes—
155SL(2,ℤ/4)—21no—
156SL(2,ℤ/4)—62no—
157SL(2,ℤ/4)—122no—
158C3×C4⋊C4—62no—
159C3×C22⋊C4—62no—
160C3×C22⋊C4—61no—
161C3×C22⋊C4—61no—
162C3×C42—62no—
163#217—64no—
164#217—62no—
165#217—62no—
166#217—62no—
167#217—62no—
168(C2×D3)⋊C4—122no—
169C4⋊Dic3—62no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—121no—
173C4⋊Dic3—241no—
174Dic3⋊C4—64no—
175Dic3⋊C4—62no—
176Dic3⋊C4—62no—
177Dic3⋊C4—122no—
178C22×C32—41no—
179C2×C3×D3—42no—
180C2×C3×D3—81no—
181C2×C3×D3—81no—
182C3×A4C22⋊Q811yescharacteristic, complemented
183C32×C4—42no—
184C3⋊Dic3—82no—
185C3×Dic3—82no—
186C3×Dic3—81no—
187C22×Q8—62no—
188C22×Q8—91no—
189C22×Q8—91no—
190C22×Q8—92no—
191C22×Q8—182no—
192C22×D4—31no—
193C23×C4—31no—
194C23×C4—32no—
195C23×C4D3211yescharacteristic
196C22⋊Q8—181no—
197C22⋊Q8—182no—
198C22⋊Q8—184no—
199C22⋊Q8—181no—
200C4×D4—62no—
201C4×D4—62no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—91no—
204C2×C4⋊C4—91no—
205C2×C4⋊C4—92no—
206C2×C4⋊C4—92no—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—92no—
210C2×C22⋊C4—92no—
211C2×C22⋊C4—92no—
212C2×C42—31no—
213#95—92no—
214#95—92no—
215#95—92no—
216#95—92no—
217#95—92no—
218C23×C3—31no—
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—31no—
222C23×C3—32no—
223C23×C3—61no—
224C23×C3—61no—
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3D3×Q811yescharacteristic
230C23×C3#23911yescharacteristic
231C23×C3C2×#6711yescharacteristic
232C22×D3—41no—
233C2×A4—21no—
234C2×A4—21no—
235C2×A4—21no—
236C2×A4C2×Dic611yescharacteristic
237C2×A4#23511yescharacteristic
238C2×A4C2×#6711yescharacteristic
239S4—22no—
240C3×Q8—242no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—62no—
245C2×C3×C4—31no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—32no—
250C2×C3×C4—41no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—81no—
261C2×C3×C4—122no—
262C2×C3×C4—122no—
263C2×C3×C4—122no—
264C2×C3×C4C2×S411yescharacteristic
265C2×Dic3—34no—
266C2×Dic3—32no—
267C2×Dic3—41no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—122no—
278C2×Dic3—242no—
279C2×Dic3C2×S412yes—
280C4×D3—82no—
281Dic6—22no—
282Dic6—64no—
283Dic6—64no—
284Dic6—62no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C3×D3—82no—
289C24—31no—
290C24C2×D3211yescharacteristic
291C2×Q8—31no—
292C2×Q8—92no—
293C2×Q8—94no—
294C2×Q8—91no—
295C2×Q8—181no—
296C2×Q8—182no—
297C2×Q8—182no—
298C2×Q8—182no—
299C2×Q8—184no—
300C2×Q8—184no—
301C2×D4—32no—
302C2×D4—32no—
303C2×D4—61no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4—61no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—62no—
314C22×C4—61no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—62no—
318C22×C4—92no—
319C22×C4—92no—
320C22×C4—92no—
321C22×C4—92no—
322C22×C4—182no—
323C22×C4#62512yes—
324C4⋊C4—62no—
325C4⋊C4—182no—
326C4⋊C4—181no—
327C4⋊C4—182no—
328C4⋊C4—184no—
329C4⋊C4—182no—
330C4⋊C4—181no—
331C22⋊C4—62no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—182no—
335C22⋊C4—184no—
336C22⋊C4—182no—
337C22⋊C4—182no—
338C22⋊C4—182no—
339C42—62no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—41no—
350C22×C3—61no—
351C22×C3—62no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—81no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3#94611yescharacteristic, complemented
360C22×C3C22×S411yescharacteristic
361C2×D3—42no—
362C2×D3—81no—
363C2×D3—81no—
364A4—21no—
365A4#92411yescharacteristic, complemented
366C3×C4—32no—
367C3×C4—42no—
368C3×C4—62no—
369C3×C4—82no—
370C3×C4—121no—
371C3×C4—121no—
372C3×C4—242no—
373C3×C4GL(2,ℤ/4)12yes—
374Dic3—22no—
375Dic3—62no—
376Dic3—81no—
377Dic3—122no—
378Dic3—122no—
379Dic3—242no—
380Dic3—482no—
381C32—41noSylow
382C23—31no—
383C23—31no—
384C23—31no—
385C23—31no—
386C23—32no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23D3×Dic611yescharacteristic
394C23#379911yescharacteristic
395C23C2×#62511yescharacteristic
396Q8—62no—
397Q8—184no—
398Q8—184no—
399Q8—182no—
400D4—62no—
401D4—62no—
402D4—62no—
403D4—62no—
404C2×C4—31no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—32no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—94no—
420C2×C4—92no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4D3×S411yescharacteristic
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—81no—
441C2×C3—81no—
442C2×C3—81no—
443C2×C3—122no—
444C2×C3Q8×S411yescharacteristic
445C2×C3#591911yescharacteristic
446C2×C3C2×GL(2,ℤ/4)11yescharacteristic
447D3—82no—
448C22—31no—
449C22—31no—
450C22—31no—
451C22—31no—
452C22—31no—
453C22—31no—
454C22—31no—
455C22—31no—
456C22—31no—
457C22—61no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22#6385611yescharacteristic, complemented
466C22C2×D3×S411yescharacteristic
467C4—32no—
468C4—62no—
469C4—62no—
470C4—121no—
471C4—121no—
472C4—182no—
473C4—362no—
474C4—362no—
475C4#6420912yes—
476C3—41no—
477C3—81no—
478C3#8550611yescharacteristic, complemented
479C2—31no—
480C2—31no—
481C2—31no—
482C2—31no—
483C2—122no—
484C2Dic6×S411yescharacteristic
485C2#1059642111yescharacteristic
486C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C4D3×S4
hypocenterC3×A4C22⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×A4×Dic6C2semidirect
3. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
5. #10593238C2semidirect
16. C23×Dic6D3semidirect
36. C2×C3×S4Q8semidirect
116. C2×S4Dic6semidirect
151. C2×Dic6S4semidirect
182. C3×A4C22⋊Q8semidirect
359. C22×C3#946semidirect
365. A4#924semidirect
465. C22#63856semidirect
478. C3#85506semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×A4×Dic62
C2×C3×C4×S42
C4⋊(C3⋊SL(2,ℤ/4))2
(C2×S4)⋊Dic32
#105932382
#855063
#823493
#638564